By G. Iooss (Eds.)
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Extra info for Bifurcation of Maps and Applications
Let t h e map . p = 1,2,. , n - 1 F P be of c l a s s ,n25 . Ck ,k > - n + l , and assume t h a t :1 # Then t h e r o t a t i o n number p(p) of f P 1 is a continuous f u n c t i o n o f p i n t h e neighborhood of 0 where y& e x i s t s , and n-2 i s a polynomial i n p of degree p(p) = Ol(p) + O( p [ y, , where 0 1(P) [91 I ( i n t e g e r p a r t of -_ 2 n-3 1. , Bifurcation of Maps and Applications 50 Proof. Let us f i r s t assume t h a t kT3 = 1 , p 2 1 , k 12p+4 . 32P+l(rp,p) A cos[2n(2p+3)cp] + B sin[2n(2p+3)tp] Let us note 2 then a2(o)rO(p) ro(w) t h e unique + p R e X1 > 0 .
In The main t o o l of t h i s paragraph i s t h e following Lemma 3 . 7 1 Assume t h a t t h e homeomorphism f where 0 > 0 t a k e s t h e form , and g i s uniformly bounded when r o t a t i o n number of p(p) P ( d = e(P) Proof. P + f iL /pI i s small. Then t h e satisfies o(lPla) - This follows d i r e c t l y from t h e formula We can now prove Theorem 3 . Let t h e map . p = 1,2,. , n - 1 F P be of c l a s s ,n25 . Ck ,k > - n + l , and assume t h a t :1 # Then t h e r o t a t i o n number p(p) of f P 1 is a continuous f u n c t i o n o f p i n t h e neighborhood of 0 where y& e x i s t s , and n-2 i s a polynomial i n p of degree p(p) = Ol(p) + O( p [ y, , where 0 1(P) [91 I ( i n t e g e r p a r t of -_ 2 n-3 1.
J1Re Now A1+ O(e3/*) , perhaps smaller, so t h a t < 1 i n (30) , and 5 l a r g e enough t o have 6E U' 1 ' The map u -t 6 U; topology i n i s a contraction i n . * U; , by ( 2 6 ) , i f we choose t h e Co This space i s complete, hence t h i s ensures t h e existence of a unique fixed point i n (34) E u = Z(U*) * : Hopf b i f u r c a t i o n i n lR2 For L >1 ) Ip , pi constants and Iv$ ) . < pl progressive a s f o r ! a2 ~ IM,l a"; a$a$ we use an equation s i m i l a r t o (27) f o r where t h e non-written terms a r e f u n c t i o n of lp'l 41 5 ap'u P; P; .
Bifurcation of Maps and Applications by G. Iooss (Eds.)